Problem: Simplify the following expression and state the condition under which the simplification is valid. $y = \dfrac{-2a^2 - 8a + 90}{-8a^3 + 200a}$
Solution: First factor out the greatest common factors in the numerator and in the denominator. $ y = \dfrac {-2(a^2 + 4a - 45)} {-8a(a^2 - 25)} $ $ y = \dfrac{2}{8a} \cdot \dfrac{a^2 + 4a - 45}{a^2 - 25} $ Simplify: $ y = \dfrac{1}{4a} \cdot \dfrac{a^2 + 4a - 45}{a^2 - 25}$ Next factor the numerator and denominator. $ y = \dfrac{1}{4a} \cdot \dfrac{(a - 5)(a + 9)}{(a - 5)(a + 5)}$ Assuming $a \neq 5$ , we can cancel the $a - 5$ $ y = \dfrac{1}{4a} \cdot \dfrac{a + 9}{a + 5}$ Therefore: $ y = \dfrac{ a + 9 }{ 4a(a + 5)}$, $a \neq 5$